The characteristic frequency and bandwidth of the random response to parametric excitation may be influenced by the excitation processes. It is demonstrated that the effective stiffness and damping properties can be expressed as conditional mean values for given displacement and energy levels, respectively. These properties are used to describe the response in terms of its probability density function and its spectral density function. An example demonstrates the accuracy in predicting change of frequency and damping of a parametrically excited oscillator, and another example extends the method to a self-excited oscillator with domains of negative damping.
Issue Section:
Technical Papers
1.
Gardiner, C. W., 1985, Handbook of Stochastic Methods, Springer-Verlag, Berlin.
2.
Lin, Y. K., and Cai, G. Q., 1995, Probabilistic Structural Dynamics, McGraw-Hill, New York.
3.
Caughey, T. K., 1971, “Nonlinear Theory of Random Vibrations,” Advances in Applied Mechanics, Vol. 11, C. S. Yih, ed., Academic Press, New York, pp. 209–253.
4.
Caughey
, T. K.
, 1986
, “On the Response of Non-linear Oscillators to Stochastic Excitation
,” Probab. Eng. Mech.
, 1
, pp. 2
–4
.5.
Roberts
, J. B.
, 1978
, “The Energy Envelope of a Randomly Excited Non-linear Oscillator
,” J. Sound Vib.
, 60
, pp. 177
–185
.6.
Roberts, J. B., 1983, “Energy Methods for Non-linear Systems With Non-white Excitation,” IUTAM Symposium on Random Vibrations and Reliability, K. Hennig, ed., Akademie-Verlag, Berlin, pp. 285–294.
7.
Wong
, E.
, and Zakai
, M.
, 1965
, “On the Relation Between Ordinary and Stochastic Differential Equations
,” Int. J. Eng. Sci.
, 3
, pp. 213
–229
.8.
Zhu
, W.
, and Lin
, Y. K.
, 1991
, “Stochastic Averaging of Energy Envelope
,” J. Eng. Mech.
, 117
, pp. 1890
–1905
.9.
Dimentberg
, M. F.
, 1982
, “An Exact Solution to a Certain Non-linear Random Vibration Problem
,” Int. J. Non-Linear Mech.
, 17
, pp. 231
–236
.10.
Lin
, Y. K.
, and Cai
, G. Q.
, 1988
, “Exact Stationary-Response Solution for Second-Order Nonlinear Systems Under Parametric and External White-Noise Excitations: Part II
,” ASME J. Appl. Mech.
, 55
, pp. 702
–705
.11.
Cai
, G. Q.
, and Lin
, Y. K.
, 1988
, “A New Approximate Solution Technique for Randomly Excited Non-linear Oscillators
,” Int. J. Non-Linear Mech.
, 23
, pp. 409
–420
.12.
Cai
, G. Q.
, Lin
, Y. K.
, and Elishakoff
, I.
, 1992
, “A New Approximate Solution Technique for Randomly Excited Non-linear Oscillators—II
,” Int. J. Non-Linear Mech.
, 27
, pp. 969
–979
.13.
Cai
, G. Q.
, and Lin
, Y. K.
, 1997
, “Comparing Quasi-Conservative Averaging and Dissipation Energy Balancing Methods in Non-linear Random Vibration
,” Int. J. Non-Linear Mech.
, 32
, pp. 121
–126
.14.
Stratonovich, R. L., 1963, Topics in the Theory of Random Noise, Vols. 1 and 2, Gordon and Breach, New York.
15.
Cai
, G. Q.
, and Lin
, Y. K.
, 1997
, “Response Spectral Densities of Strongly Nonlinear Systems Under Random Excitation
,” Probabilistic Engineering Mechanics
, 12
, pp. 41
–47
.16.
Krenk
, S.
, and Roberts
, J. B.
, 1999
, “Local Similarity in Non-linear Random Vibration
,” ASME J. Appl. Mech.
, 66
, pp. 225
–235
.17.
Krenk, S., 1999, “Energy and Spectral Density in Non-linear Random Response,” Stochastic Structural Dynamics, B. F. Spencer and E. A. Johnson, eds., Balkema, Rotterdam, pp. 43–51.
18.
Yong
, Y.
, and Lin
, Y. K.
, 1987
, “Exact Stationary-Response Solution for Second-Order Nonlinear Systems Under Parametric and External White-Noise Excitations
,” ASME J. Appl. Mech.
, 54
, pp. 414
–418
.19.
Lin
, Y. K.
, and Cai
, G. Q.
, 1988
, “Equivalent Stochastic Systems
,” ASME J. Appl. Mech.
, 55
, pp. 918
–922
.20.
Fogli
, M.
, Bresollette
, P.
, and Bernard
, P.
, 1996
, “The Dynamics of a Stochastic Oscillator With Impacts
,” Eur. J. Mech. A/Solids
, 15
, pp. 213
–241
.21.
Ru¨dinger, F., and Krenk, S., 2001, “Stochastic Analysis of Self-Induced Vibrations,” submitted to Meccanica, for publication.
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